1. In this question, we study the hypothesis class of Boolean conjunctions defined as follows. The instance space is X ={0,1}d and the label set is Y ={0,1}. A literal over the variables x1, . . ., xd is a simple Boolean function that takes the form f (x)= xi, for some i ∈ [d], or f (x) = 1−xi for some i ∈ [d]. We use the notation .xi as a shorthand
for 1− xi . A conjunction is any product of literals. In Boolean logic, the product is denoted using the ∧ sign. For example, the function h(x) = x1 · (1− x2) is written as x1 ∧ .x2. We consider the hypothesis class of all conjunctions of literals over the d variables. The empty conjunction is interpreted as the all-positive hypothesis (namely, the function that returns h(x) = 1 for all x). The conjunction x1 ∧ .x1 (and similarly any conjunction involving a literal and its negation) is allowed and interpreted as the all-negative hypothesis (namely, the conjunction that returns h(x) = 0 for all x). We assume realizability: Namely, we assume that there exists a Boolean conjunction that generates the labels. Thus, each example (x, y) ∈ X × Y consists of an assignment to the d Boolean variables x1, . . ., xd , and its truth value (0 for false and 1 for true). For instance, let d = 3 and suppose that the true conjunction is x1 ∧ .x2. Then, the training set S might contain the following instances: ((1,1, 1), 0), ((1,0, 1), 1), ((0,1, 0),0)((1,0, 0), 1).
Prove that the hypothesis class of all conjunctions over d variables is PAC learnable and bound its sample complexity. Propose an algorithm that implements the ERM rule, whose runtime is polynomial in d ·m.
